For the dynamics of disk galaxies the basic situation to be
considered is that of a mean field potential
that is stationary and
axisymmetric around the z-axis. Note that such potential is very
different
from the Keplerian potential generated by a point mass located at
r = 0. In the
equatorial plane defined by z = 0, the calculation of orbits is
reduced to a one-dimensional problem by introducing an effective potential

(13.8)

so that the energy integral can be written as

(13.9)

Figure 13.2. Sketch of the effective
potential for equatorial orbits in an axisymmetric field.

Thus the radial momentum (in our case this is identified with the
radial velocity) can be expressed as a function of r and of the
integrals of
the motion E and J, with J the specific angular
momentum. For a large class of potentials, the function
eff exhibits
one minimum at r = r0 (see
Fig. 13.2), which
identifies the radius of circular orbits
with angular momentum J. If we take J > 0, and define

(13.10)

the guiding center radius is related to the specific angular
momentum by

(13.11)

which is generally one-to-one; in order for
eff to
exhibit a minimum at r0, the function
J = J(r0) defined by Eq. (11) must be
monotonically increasing.
Typically, for a given value of J bound orbits are associated with
energies in the range E0 < E < 0, with

(13.12)

the minimum energy which corresponds to the circular orbit. In the
radial coordinate the motion is periodic and takes place between two turning
points
rin(E, J) < r0 <
rout(E, J). A radial action variable can
thus be set

(13.13)

with the property

(13.14)

Here the radial frequency is defined as
r =
2 /
r,
with the bounce time given by

(13.15)

In turn, the angular frequency is defined by

(13.16)

Orbits are closed (in the inertial frame of reference) if the ratio
between the two frequencies is rational.